Finite Group Databasebeta

← #35227298

Subgroups — #35227298

Order 20864.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782393C211yesmaximal, characteristic, complemented
2#23782029C211yesmaximal, characteristic
3#23782024C211yesmaximal, characteristic, complemented
4#23782019C211yesmaximal, characteristic, complemented
5#23781972C211yesmaximal, characteristic, complemented
6#23781965C211yesmaximal, characteristic, complemented
7#23781955C211yesmaximal, characteristic, complemented
8#23781884C211yesmaximal, characteristic, complemented
9#23781884C211yesmaximal, characteristic, complemented
10#23781862C211yesmaximal, characteristic
11#23781796C211yesmaximal, characteristic, complemented
12#23781770C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14C4×#14579687C211yesmaximal, characteristic, complemented
15#559×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104735C2211yescharacteristic
20#19104733—21no—
21#19104727C2211yescharacteristic
22#19104727C2211yescharacteristic
23#19104725—21no—
24#19104725—21no—
25C4×Dic326C2211yescharacteristic
26C4×Dic326C2211yescharacteristic
27C4×Dic326C2211yescharacteristic
28#19104700—21no—
29#19104668C2211yescharacteristic
30#19104666C2211yescharacteristic
31#19104666C2211yescharacteristic
32C4⋊(C163⋊C8)C2211yescharacteristic
33C4⋊(C163⋊C8)C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104659C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104658C2211yescharacteristic
39#19104656C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104656C2211yescharacteristic
45#19104646C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48#127×C163C2211yescharacteristic
49#124×C163—21no—
50#124×C163C2211yescharacteristic
51#124×C163C2211yescharacteristic
52#123×C163—21no—
53#123×C163—21no—
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4×D4×C163—21no—
57C4×D4×C163C2211yescharacteristic
58C4×D4×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61#14579689—22no—
62#14579687—22no—
63#14579687—22no—
64C2×C163⋊C8C2311yescharacteristic
65C2×C163⋊C8C2311yescharacteristic
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C4⋊Dic163—21no—
69C4⋊Dic163C2311yescharacteristic
70C4⋊Dic163C2311yescharacteristic
71C4⋊Dic163C2311yescharacteristic
72Dic163⋊C4—21no—
73Dic163⋊C4—21no—
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C4×Dic163—21no—
80C4×Dic163—21no—
81C4×Dic163—21no—
82C2×Q8×C163D411yescharacteristic
83C2×Q8×C163C2311yescharacteristic
84C2×D4×C163—21no—
85C2×D4×C163—21no—
86C2×D4×C163C2311yescharacteristic
87C2×D4×C163C2311yescharacteristic
88C22×C4×C163—21no—
89C22×C4×C163—21no—
90C4⋊C4×C163—21no—
91C4⋊C4×C163D411yescharacteristic
92C4⋊C4×C163D411yescharacteristic
93C4⋊C4×C163D411yescharacteristic
94C4⋊C4×C163C2311yescharacteristic
95C22⋊C4×C163—21no—
96C22⋊C4×C163—21no—
97C22⋊C4×C163—21no—
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C42×C163—21no—
102C42×C163C2311yescharacteristic
103C42×C163C2311yescharacteristic
104C42×C163C2311yescharacteristic
105Dic326—21no—
106Dic326—42no—
107Dic326#4112yes—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C163⋊C8—21no—
113C163⋊C8—21no—
114C163⋊C8—21no—
115C163⋊C8—21no—
116C23×C163—21no—
117C23×C163—21no—
118Q8×C163—21no—
119Q8×C163—21no—
120Q8×C163—41no—
121D4×C163—21no—
122D4×C163—21no—
123D4×C163—21no—
124D4×C163—21no—
125D4×C163—42no—
126D4×C163—81no—
127C2×C4×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—81no—
130C2×C4×C163—81no—
131C2×C4×C163C2×D411yescharacteristic
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2411yescharacteristic
138C22×C163—41no—
139C22×C163—41no—
140C22×C163—41no—
141C22×C163—41no—
142C22×C163—41no—
143C22×C163—41no—
144C22×C163C22×D411yescharacteristic
145C4×C163—21no—
146C4×C163—21no—
147C4×C163—21no—
148C4×C163—21no—
149C4×C163—21no—
150C4×C163—21no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163C2×#4111yescharacteristic
154C4×C1632−1+411yescharacteristic
155Dic163—22no—
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159C2×C163—81no—
160C2×C163—81no—
161C2×C163#54811yescharacteristic
162C2×C163#57511yescharacteristic
163C2×C163#57711yescharacteristic
164C163#337111yescharacteristic, complemented, Sylow
165#3371—1631nomaximal, Sylow
166#559D16311yescharacteristic
167#557—1631no—
168#490—1631no—
169#489—1631no—
170#488—1631no—
171#479—1631no—
172#478—1631no—
173#476—1631no—
174#464—1631no—
175#464—1631no—
176#460—1631no—
177#446—1631no—
178#441—1631no—
179#440—1631no—
180C4×SD16—1631no—
181C2×Q16—1631no—
182C2×SD16—1631no—
183C2×SD16—1631no—
184C4⋊Q8—3261no—
185C4⋊Q8—1631no—
186#127C2×D16311yescharacteristic
187#125—3261no—
188#125—3261no—
189#125—1631no—
190#125—1631no—
191#124—21no—
192#124C2×D16311yescharacteristic
193#124C2×D16311yescharacteristic
194#123—21no—
195#123—21no—
196C4×Q8—3261no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200C4×Q8C2×D16311yescharacteristic
201C4×Q8C2×D16311yescharacteristic
202C4×D4—21no—
203C4×D4C2×D16311yescharacteristic
204C4×D4C2×D16311yescharacteristic
205#107—1631no—
206#106—1631no—
207#106—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210C4⋊C8—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#102—1631no—
216#102—1631no—
217#102—1631no—
218#102—1631no—
219#102—1631no—
220#102—1631no—
221#97—1631no—
222#97—1631no—
223C4×C8—1631no—
224C2×Q8—3261no—
225C2×Q8—1631no—
226C2×Q8C22⋊D16311yescharacteristic
227C2×Q8C22×D16311yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C22×D16311yescharacteristic
231C2×D4C22×D16311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234Q16—3262no—
235SD16—3262no—
236SD16—3262no—
237C2×C8—1631no—
238C2×C8—1631no—
239C2×C8—1631no—
240C2×C8—1631no—
241C4⋊C4—21no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—1631no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4C22⋊D16311yescharacteristic
254C4⋊C4C22⋊D16311yescharacteristic
255C4⋊C4C22⋊D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C42—21no—
264C42—3261no—
265C42—3261no—
266C42—3261no—
267C42C22×D16311yescharacteristic
268C42C22×D16311yescharacteristic
269C42C22×D16311yescharacteristic
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—41no—
275Q8—3261no—
276Q8—6522no—
277Q8—1632no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4C2×C22⋊D16311yescharacteristic
293C2×C4C2×C22⋊D16311yescharacteristic
294C2×C4C2×C22⋊D16311yescharacteristic
295C2×C4C2×C22⋊D16311yescharacteristic
296C2×C4C2×C22⋊D16311yescharacteristic
297C2×C4C2×C22⋊D16311yescharacteristic
298C2×C4C23×D16311yescharacteristic
299C8—3261no—
300C8—3261no—
301C8—3261no—
302C8—3261no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C22×C22⋊D16311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—3262no—
319C4—6521no—
320C4—6521no—
321C4—6521no—
322C4#41×D16311yescharacteristic
323C4#1910478211yescharacteristic
324C2—81no—
325C2—81no—
326C2#2378232211yescharacteristic
327C2#2378246311yescharacteristic
328C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#559×C163C2
socleC22×C163C22×D4
prime radical#127×C163C22
hypercenter#559D163
hypocenterC163#3371
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23782393C2semidirect
3. #23782024C2semidirect
4. #23782019C2semidirect
5. #23781972C2semidirect
6. #23781965C2semidirect
7. #23781955C2semidirect
8. #23781884C2semidirect
9. #23781884C2semidirect
11. #23781796C2semidirect
12. #23781770C2semidirect
13. #23781765C2semidirect
14. C4×#14579687C2semidirect
164. C163#3371semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237823932
#237820292
#237820242
#237820192
#237819722
#237819652
#237819552
#237818842
#237818622
#237817962
#237817702
#237817652
C4×#145796872
#559×C1632
#3371163